I have obtained the expressions for the electric (scalar) potential and magnetic (Vector) potential and substituted them in as the V(r,t) that appears in Schrodinger's equation. The trouble I'm having at the moment is that would V(r,t) just be a sum of electric and magnetic potential? Also, the question also mentions I should introduce a gauge to simply the equation, but I'm not sure how this gauge transformation would look like.

To reformulate what Gokul said, what is the hamiltonian for a nonrelativistic electrically charged massive particle in a em field. Having got that, obtaining the quantum hamiltonian is a piece of cake.

Sorry I haven't replied in a few days. But, i've been flat out with other subjects. OK, back to this problem, I have found a solution to this where they used Lagrangian-Hamiltonian to formulate the solution. And it seems like they obtain a Hamiltonian of [tex]H = q\phi + \frac{1}{2m} \left[ p - qA\right]^{2}[/tex], where q is the charged particle, m is the mass, p is the momentum and A is the vector potential. I'm stepping through each of the steps since I don't want to just hand the answer in like that without understanding it. I'm sorta stuck on the derivation at part [IV-2], where they take the total time derivative of the vector potential. Can someone shed some light on why they do this? In the question specified above the E field and B field are constant, can I still consider a time varying vector potential to solve for constant crossed E and B field?

I've got the generalised quantum Hamiltonian for the case above. Now, I want to obtain the relevant scalar/vector potentials given the E/B fields specified above. I have solved the vector potential expression, and related all the corresponding B and A components to their respected Cartesian components.

Now the problem i'm having is, how can i recombine all these equations together to obtain a value for A, where A is the vector potential as a function of x,y,z and t (time). This is needed since the expression of the Hamiltonian needs A.
Anyone have some tips for this solution?

I guess it's not the Landau gauge that you want, but probably a close extension of it....sorry. I forgot about the E-field in the problem. To cut to the chase, what happens if you choose say, A = (zB,0,0)?

Taking the Curl of (zB, 0, 0) it returns B = (0, B, 0), which agrees with the B field specified. But also, we can take the curl of (0, 0, -xB) and this yields B = (0, B, 0). Is the zB one correct or -xB one correct or both are correct? I'm thinking it has something to do with the symmetry of the system.

However, if I do proceed with using A = (zB, 0, 0) and the scalar potential [tex]\phi = -zE [/tex]. (Note: the time partial derivative term in the expression relating E and the scalar potential is cancelled out since A isn't not dependent on time since B is constant, there [tex]E = -\nabla \phi [/tex])

I think it does get eliminated. With [itex] \vec{\nabla} . \vec{A}\right) = 0 [/itex], you have [itex]\vec{A}.\vec{\nabla} \psi + \vec{\nabla}.\vec{A}\psi = 2\vec{A}.\vec{\nabla} \psi~~ [/itex] ... or am I making some mistake too?

Yep, you get [tex] 2 \vec{A} . \nabla [/tex]. Then this will cancel out, just wondering if my final equation seems logical? With all the steps taken into account, also how about if A was equal to A = (0, 0, -xB) which still yields B = (0, B, 0)

Now the problem i'm having is that i split this up further to the individual components (i.e. x, y, z), am I allowed to disregard the terms that aren't dependent on the other components. Say for the x dependence I can eliminate the constants 3rd and 4th term in the square brackets but still keep the 2nd term since there is a partial derivative with respect to x. If i do this i get these 3 equations:

This doesn't look like a 1-D problem to me, I still see the partial derivative with respect x in the z-cartesian equation. Can anyone give me any hints on how to reduce this any further, or if i've gone down totally wrong pathway.

I have exactly the same problem, with the same fields. Determining the Schrödinger equation is relatively straight foreword. Now the problem is to solve the equation to determine the wavefunction \Psy. I have a hint to separate variables to reduce it to a one dimensional problem. It is straight foreword to separate out the spatial and temporal parts, but the spatial equation then looks horrible. Can anyone help?
Essentially the equation to be solved is: